Basic Math Examples

${(2 t + \frac{1}{t})}^{7}$

Step 1

Use the Binomial Theorem.

${(2 t)}^{7} + 7 {(2 t)}^{6} \frac{1}{t} + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $2 t$ .

$2^{7} t^{7} + 7 {(2 t)}^{6} \frac{1}{t} + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.2

Raise $2$ to the power of $7$ .

$128 t^{7} + 7 {(2 t)}^{6} \frac{1}{t} + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.3

Apply the product rule to $2 t$ .

$128 t^{7} + 7 (2^{6} t^{6}) \frac{1}{t} + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.4

Cancel the common factor of $t$ .

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Step 2.4.1

Factor $t$ out of $7 (2^{6} t^{6})$ .

$128 t^{7} + t (7 (2^{6} t^{5})) \frac{1}{t} + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.4.2

Cancel the common factor.

Step 2.4.3

Rewrite the expression.

$128 t^{7} + 7 (2^{6} t^{5}) + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.5

Raise $2$ to the power of $6$ .

$128 t^{7} + 7 (64 t^{5}) + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.6

Multiply $64$ by $7$ .

$128 t^{7} + 448 t^{5} + 21 {(2 t)}^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.7

Apply the product rule to $2 t$ .

$128 t^{7} + 448 t^{5} + 21 (2^{5} t^{5}) {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.8

Raise $2$ to the power of $5$ .

$128 t^{7} + 448 t^{5} + 21 (32 t^{5}) {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.9

Multiply $32$ by $21$ .

$128 t^{7} + 448 t^{5} + 672 t^{5} {(\frac{1}{t})}^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.10

Apply the product rule to $\frac{1}{t}$ .

$128 t^{7} + 448 t^{5} + 672 t^{5} \frac{1^{2}}{t^{2}} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.11

Cancel the common factor of $t^{2}$ .

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Step 2.11.1

Factor $t^{2}$ out of $672 t^{5}$ .

$128 t^{7} + 448 t^{5} + t^{2} (672 t^{3}) \frac{1^{2}}{t^{2}} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.11.2

Cancel the common factor.

Step 2.11.3

Rewrite the expression.

$128 t^{7} + 448 t^{5} + 672 t^{3} \cdot 1^{2} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.12

One to any power is one.

$128 t^{7} + 448 t^{5} + 672 t^{3} \cdot 1 + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.13

Multiply $672$ by $1$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 35 {(2 t)}^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.14

Apply the product rule to $2 t$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 35 (2^{4} t^{4}) {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.15

Raise $2$ to the power of $4$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 35 (16 t^{4}) {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.16

Multiply $16$ by $35$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t^{4} {(\frac{1}{t})}^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.17

Apply the product rule to $\frac{1}{t}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t^{4} \frac{1^{3}}{t^{3}} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.18

Cancel the common factor of $t^{3}$ .

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Step 2.18.1

Factor $t^{3}$ out of $560 t^{4}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + t^{3} (560 t) \frac{1^{3}}{t^{3}} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.18.2

Cancel the common factor.

$128 t^{7} + 448 t^{5} + 672 t^{3} + t^{3} (560 t) \frac{1^{3}}{t^{3}} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.18.3

Rewrite the expression.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t \cdot 1^{3} + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.19

One to any power is one.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t \cdot 1 + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.20

Multiply $560$ by $1$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + 35 {(2 t)}^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.21

Apply the product rule to $2 t$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + 35 (2^{3} t^{3}) {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.22

Raise $2$ to the power of $3$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + 35 (8 t^{3}) {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.23

Multiply $8$ by $35$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + 280 t^{3} {(\frac{1}{t})}^{4} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.24

Apply the product rule to $\frac{1}{t}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + 280 t^{3} \frac{1^{4}}{t^{4}} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.25

Cancel the common factor of $t^{3}$ .

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Step 2.25.1

Factor $t^{3}$ out of $280 t^{3}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + t^{3} \cdot 280 \frac{1^{4}}{t^{4}} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.25.2

Factor $t^{3}$ out of $t^{4}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + t^{3} \cdot 280 \frac{1^{4}}{t^{3} t} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.25.3

Cancel the common factor.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + t^{3} \cdot 280 \frac{1^{4}}{t^{3} t} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.25.4

Rewrite the expression.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + 280 \frac{1^{4}}{t} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.26

Combine $280$ and $\frac{1^{4}}{t}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280 \cdot 1^{4}}{t} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.27

One to any power is one.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280 \cdot 1}{t} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.28

Multiply $280$ by $1$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + 21 {(2 t)}^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.29

Apply the product rule to $2 t$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + 21 (2^{2} t^{2}) {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.30

Raise $2$ to the power of $2$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + 21 (4 t^{2}) {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.31

Multiply $4$ by $21$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + 84 t^{2} {(\frac{1}{t})}^{5} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.32

Apply the product rule to $\frac{1}{t}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + 84 t^{2} \frac{1^{5}}{t^{5}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.33

Cancel the common factor of $t^{2}$ .

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Step 2.33.1

Factor $t^{2}$ out of $84 t^{2}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + t^{2} \cdot 84 \frac{1^{5}}{t^{5}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.33.2

Factor $t^{2}$ out of $t^{5}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + t^{2} \cdot 84 \frac{1^{5}}{t^{2} t^{3}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.33.3

Cancel the common factor.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + t^{2} \cdot 84 \frac{1^{5}}{t^{2} t^{3}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.33.4

Rewrite the expression.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + 84 \frac{1^{5}}{t^{3}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.34

Combine $84$ and $\frac{1^{5}}{t^{3}}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84 \cdot 1^{5}}{t^{3}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.35

One to any power is one.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84 \cdot 1}{t^{3}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.36

Multiply $84$ by $1$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + 7 (2 t) {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.37

Multiply $2$ by $7$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + 14 t {(\frac{1}{t})}^{6} + {(\frac{1}{t})}^{7}$

Step 2.38

Apply the product rule to $\frac{1}{t}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + 14 t \frac{1^{6}}{t^{6}} + {(\frac{1}{t})}^{7}$

Step 2.39

Cancel the common factor of $t$ .

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Step 2.39.1

Factor $t$ out of $14 t$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + t \cdot 14 \frac{1^{6}}{t^{6}} + {(\frac{1}{t})}^{7}$

Step 2.39.2

Factor $t$ out of $t^{6}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + t \cdot 14 \frac{1^{6}}{t \cdot t^{5}} + {(\frac{1}{t})}^{7}$

Step 2.39.3

Cancel the common factor.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + t \cdot 14 \frac{1^{6}}{t \cdot t^{5}} + {(\frac{1}{t})}^{7}$

Step 2.39.4

Rewrite the expression.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + 14 \frac{1^{6}}{t^{5}} + {(\frac{1}{t})}^{7}$

Step 2.40

Combine $14$ and $\frac{1^{6}}{t^{5}}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + \frac{14 \cdot 1^{6}}{t^{5}} + {(\frac{1}{t})}^{7}$

Step 2.41

One to any power is one.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + \frac{14 \cdot 1}{t^{5}} + {(\frac{1}{t})}^{7}$

Step 2.42

Multiply $14$ by $1$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + \frac{14}{t^{5}} + {(\frac{1}{t})}^{7}$

Step 2.43

Apply the product rule to $\frac{1}{t}$ .

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + \frac{14}{t^{5}} + \frac{1^{7}}{t^{7}}$

Step 2.44

One to any power is one.

$128 t^{7} + 448 t^{5} + 672 t^{3} + 560 t + \frac{280}{t} + \frac{84}{t^{3}} + \frac{14}{t^{5}} + \frac{1}{t^{7}}$